Appendix B: Multivariable Calculus

Transcription

1 Chapter 9 Appendix B: Multivariable Calculus 9.1 Basic Theorems of Multivariable Calculus The Euclidean spacer n is the set of n-tuples x=(x 1,..., x n )=(x i ), i= 1,...,n. The standard basis inr n is the set of vectors e i = (0,...,1,...,0) Let f : U Rbe a real valued function. The partial derivative of f at a point x with respect to x i is defined by f 1 = lim x i t 0 t [ f (x+te i) f (x)] It will be denoted by i f= f x i A function f is smooth if it has continuous partial derivatives of any order. The set of smooth functions is denoted by C (R n ). The composition of smooth functions is smooth. 227

2 228 CHAPTER 9. APPENDIX B: MULTIVARIABLE CALCULUS Let F :R n R m be a map (vector-valued function), whereµ=(1,...,m). x y=f(x)=(f 1 (x),..., F m (x))=(f µ (x)), The map F is smooth if every component F µ is smooth. The derivative DF of F at x is the m n matrix of partial derivatives called the Jacobian matrix ( ) F µ DF=. x i Chain Rule. Let F :R n R m and G :R m R k be two smooth maps. The Jacobian matrix of the composition H= G F is equal to the product of the Jacobian matrices D(G F)=DGDF. Let us denoted the coordinates ofr n,r m,r k by x i, y µ, z a respectively with i=1,...,n;µ=1,...,m; a=1,...,k, so that y=f(x), z=h(y)=g(f(x)) Then z a x i = m µ=1 z a y µ y µ x i If n=m, the determinant of the Jacobian matrix ( ) F j det DF= det x i is called the Jacobian. Let U, V R n be two open sets and F : U V be a homeomorphism. The map F is called a diffeomorphism if both F and F 1 are smooth. Inverse Function Theorem. Let F : R n R n be a smooth map. Let a R n be a point such that the Jacobian matrix DF(a) is non-degenerate at a. Then: diffgeom.tex; January 18, 2018; 9:43; p. 224

3 9.1. BASIC THEOREMS OF MULTIVARIABLE CALCULUS there exists a neighborhood U of the point a such that the set V= F(U) is open and the map F : U V has a smooth inverse F 1 : V U (that is, F is a diffeomorphism), 2. for any point x U where x=f 1 (y). (DF 1 )(y)=(df(x)) 1, Implicit Function Theorem. Let F :R r R m R m be a smooth map. Let D y F and D x F be matrices defined by ( ) ( ) F µ (x, y) F µ (x, y) D y F=, D y ν x F=, x i whereµ,ν=1,...m; i=1,...,r. Let c R m be a point inr m and W be the set of points inr r R m defined by W= F 1 (c)={(x, y) R r R m F(x, y)=c} Suppose that the set W is non-empty. Let (a, b) be a point in W, that is, F(a, b)=c, such that the matrix D y F(a, b) is not degenerate at (a, b). Then: 1. there is a neighborhood U R r of the point a and a neighborhood V R m of the point b and a unique smoth map G : U V, such that V= G(U) and for any x U, (x, G(x)) W. 2. For any point x U, D x G= (D y F) 1 D x F. Proof. By the inverse function theorem one can invert (locally) the map (with fixed x) z=f(x, y) to express the coordinates y µ in terms of the coordinates z µ, By fixing z=cwe get the map y= G(x, z). x y=g(x)= G(x, c). diffgeom.tex; January 18, 2018; 9:43; p. 225

4 230 CHAPTER 9. APPENDIX B: MULTIVARIABLE CALCULUS 9.2 Coordinates The Euclidean spacer n is the set of points x represented by n-tuples (x i )= (x 1,..., x n ), with the numbers x i called coordinates of the point x. We may introduce the coordinate functions ϕ i (x)= x i that assign the coordinates to the points. They provide a coordinate system onr n. Example. Unit sphere S 2. 3 S 2 = x R3 (x i ) 2 = 1 Problem: how to describe points on S 2? There is no single coordinate system that adequately describes S 2. i=1 One needs at least two coordinate systems. Coordinates on S 2 are described by the coordinate map ϕ :R 3 R 2 Let the sphere S 2 be centered at the origin with the north pole (0, 0, 1). Spherical coordinates are described by the coordinate map ϕ 1 (x)=θ, ϕ 2 (x)=ϕ. These coordinates are bad at both poles. Need to rotate them. Stereographic coordinates (with respect to the North (South) pole) are described by the coordinate map or where u 1 = x1 1 r, u2 = x2 1 r, v 1 = x1 1+r, v2 = x2 1+r, r= 1 (x 1 ) 2 (x 2 ) 2. diffgeom.tex; January 18, 2018; 9:43; p. 226

5 9.3. SMOOTH MAPS 231 Coordinate systems are not unique. The change of coordinates is admissible if the corresponding map is a diffeomorphism, that is, has a non-zero Jacobian. Exercises: Compute the Jacobian for the change of coordinates 1. The change from the spherical to the stereographic coordinates is given by the map u 1 = cosϕsinθ 1 cosθ = cotθ 2 cosϕ, u2 = sinϕsinθ 1 cosθ = cotθ 2 sinϕ,. 2. The change from the North pole to the South pole stereographic coordinates is given by where v 1 = u1 R 2, v2 = u2 R 2, R= (u 1 ) 2 + (u 2 ) Smooth Maps Let f : M R be a function. Let p be a point and (U,ϕ) be a chart containing p. This defines a real valued function of several variables f=f ϕ 1 :ϕ(u) R called the local representation of the function f in the cordinate chart (U,ϕ). Usually we just use the same symbol to denote it. A function f is smooth if its local representation is smooth. Let F : M N be smap between two manifolds. Let n = dim M and m = dim N. Let p M be a point in M and (U,ϕ) be a chart in M containing p. diffgeom.tex; January 18, 2018; 9:43; p. 227

6 232 CHAPTER 9. APPENDIX B: MULTIVARIABLE CALCULUS Let (V,ψ) be the chart on N containing the point F(p) as well as the set F(U). The local representation of the map F is given by the m-vector valued function of n variables F=ψ F ϕ 1 :ϕ(u) ψ(v) Let x i, i=1,...,n, be local coordinates in U and y µ,µ=1,...,m, be local coordinates in V. Then y µ = F µ (x). Again, to simplify notation we just denote this local representative by F. The map F is smooth if all its local representatives are smooth. The map F is called an immersion at a point p M if n=dim M m=dim N and the Jacobian matrix DF(p) at p has the maximal rank, rank DF(p)=n. The manifold M is called an immersed submanifold of the manifold N if the map F is an immersion at every point p M. In this case the image F(M) can be equipped by local coordinates by the inverse function theorem. The image F(M) of an immersed submanifold can have self-intersections. The map F is called an submersion at a point p M if n=dim M m=dim N and the Jacobian matrix DF(p) at p has the maximal rank, rank DF(p)=m. diffgeom.tex; January 18, 2018; 9:43; p. 228

7 9.3. SMOOTH MAPS 233 The map F is called an embeding if it is an injective immersion and the map F : M F(M) is a homeomorphism. The manifold M is called an embedded submanifold if the map F is an embedding. The image F(M) of an embedded submanifold does not have any selfintersections. Whitney Embedding Theorem. 1. Any n-dimensional topological manifold can be embedded inr 2n Any n-dimensional smooth manifold can be embedded inr 2n. Example. Klein Bottle K 2 cannot be embedded inr 3. Let M and N be two manifolds with Let F : M Nbe a smooth map. n=dim M m=dim N. A point p M is called a regular point of the map F if the Jacobian DF at p has the maximal rank (that is, F is a submersion at p ) rank DF(p)=m and a critical point if rank DF(p)<m. A point q Nis called a regular value of the map F if the inverse image F 1 (q) is either empty or every point p in the inverse image F 1 (q) is regular. Sard Theorem. The set of all regular values of a smooth map F : M N is dense in N. That is, the set of all critical values has a measure zero. We also says, that almost all values are regular. diffgeom.tex; January 18, 2018; 9:43; p. 229

8 234 CHAPTER 9. APPENDIX B: MULTIVARIABLE CALCULUS Regular Value Theorem. Let M be an n-dimensional manifold and N be a m-dimensional manifold. Suppose n m and let r = n m. Let F : M N be a smooth map. Let q N be a regular value and W M be the set defined by W= F 1 (q)={p M F(p)=q}. Then W is an embedded r-dimensional submanifold of M. Proof. This follows from the implicit function theorem. Let V be a neighborhood of q with local coordinates z µ,µ=1,...,m such that z µ (q)=c µ with some c µ. Let U = F 1 (V) and (x 1,..., x r, y 1,...,y m )=(x i, y µ ),µ=1,...,m, i= 1,...,r, be local coordinates in U. Since q is a regular value, every point p W is regular. So, the rank of the Jacobian DF(p) is maximal rank DF(p)=m Therefore, we can reorder the coordinaes (x, y) so that the m m matrix D y F(x, y) is non-degenerate. By the inverse function theorem this means that locally one can invert the map (with fixed x) z=f(x, y) to express the coordinates y µ in terms of the coordinates z µ, Now, by fixing z=cwe get the map Notice that for any x y= G(x, z). y= G(x, z)=g(x). F(x, G(x))=c, which means that the point (x, G(x)) Wand, therefore, x i provide local coordinates for W. diffgeom.tex; January 18, 2018; 9:43; p. 230